2 Specification

3 Description

nag_5d_shep_interp (e01tmc) constructs a smooth function Qx, x∈ℝ5 which interpolates a set of m scattered data points xr,fr, for r=1,2,…,m, using a modification of Shepard's method. The surface is continuous and has continuous first partial derivatives.

The basic Shepard method, which is a generalization of the two-dimensional method described in Shepard (1968), interpolates the input data with the weighted mean

Qx=∑r=1mwrxqr∑r=1mwrx,

where qr=fr, wrx=1dr2 and dr2=x-xr22.

The basic method is global in that the interpolated value at any point depends on all the data, but nag_5d_shep_interp (e01tmc) uses a modification (see Franke and Nielson (1980) and Renka (1988a)), whereby the method becomes local by adjusting each wrx to be zero outside a hypersphere with centre xr and some radius Rw. Also, to improve the performance of the basic method, each qr above is replaced by a function qrx, which is a quadratic fitted by weighted least squares to data local to xr and forced to interpolate xr,fr. In this context, a point x is defined to be local to another point if it lies within some distance Rq of it.

The efficiency of nag_5d_shep_interp (e01tmc) is enhanced by using a cell method for nearest neighbour searching due to Bentley and Friedman (1979) with a cell density of 3.

The radii Rw and Rq are chosen to be just large enough to include Nw and Nq data points, respectively, for user-supplied constants Nw and Nq. Default values of these arguments are provided, and advice on alternatives is given in Section 8.2.

nag_5d_shep_interp (e01tmc) is derived from the new implementation of QSHEP3 described by Renka (1988b). It uses the modification for five-dimensional interpolation described by Berry and Minser (1999).

Values of the interpolant Qx generated by nag_5d_shep_interp (e01tmc), and its first partial derivatives, can subsequently be evaluated for points in the domain of the data by a call to nag_5d_shep_eval (e01tnc).

On entry: the number Nw of data points that determines each radius of influence Rw, appearing in the definition of each of the weights wr, for r=1,2,…,m (see Section 3). Note that Rw is different for each weight. If nw≤0 the default value nw=min32,m-1 is used instead.

Constraint:
nw≤min50,m-1.

5:
nq – IntegerInput

On entry: the number Nq of data points to be used in the least squares fit for coefficients defining the quadratic functions qrx (see Section 3). If nq≤0 the default value nq=min50,m-1 is used instead.

6 Error Indicators and Warnings

On entry, all the data points lie on the same four-dimensional hypersurface.
No unique solution exists.

NE_DUPLICATE_NODE

There are duplicate nodes in the dataset.
x[k-1×5+i-1]=x[k-1×5+j-1],
for i=value, j=value and
k=1,2,…,5. The interpolant cannot be derived.

NE_INT

On entry, m=value.
Constraint: m≥23.

On entry, nq=value.
Constraint: nq≤0 or
nq≥20.

NE_INT_2

On entry, nq=value and m=value.
Constraint: nq≤min70,m-1.

On entry, nw=value and m=value.
Constraint: nw≤min50,m-1.

NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

7 Accuracy

On successful exit, the function generated interpolates the input data exactly and has quadratic precision. Overall accuracy of the interpolant is affected by the choice of arguments nw and nq as well as the smoothness of the function represented by the input data. Berry and Minser (1999) report on the results obtained for a set of test functions.

8 Further Comments

8.1 Timing

The time taken for a call to nag_5d_shep_interp (e01tmc) will depend in general on the distribution of the data points and on the choice of Nw and Nq parameters. If the data points are uniformly randomly distributed, then the time taken should be Om. At worst Om2 time will be required.

8.2 Choice of Nw and Nq

Default values of the arguments Nw and Nq may be selected by calling nag_5d_shep_interp (e01tmc) with nw≤0 and nq≤0. These default values may well be satisfactory for many applications.

If nondefault values are required they must be supplied to nag_5d_shep_interp (e01tmc) through positive values of nw and nq. Increasing these argument values makes the method less local. This may increase the accuracy of the resulting interpolant at the expense of increased computational cost. The default values nw=min32,m-1 and nq=min50,m-1 have been chosen on the basis of experimental results reported in Berry and Minser (1999). In these experiments the error norm was found to increase with the decrease of Nq, but to be little affected by the choice of Nw. The choice of both, directly affected the time taken by the function. For further advice on the choice of these arguments see Berry and Minser (1999).

9 Example

This program reads in a set of 30 data points and calls nag_5d_shep_interp (e01tmc) to construct an interpolating function Qx. It then calls nag_5d_shep_eval (e01tnc) to evaluate the interpolant at a set of points.

Note that this example is not typical of a realistic problem: the number of data points would normally be larger.